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Simple Circuit Extensions for XOR in PTIME

Authors Marco Carmosino , Ngu Dang , Tim Jackman

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LIPIcs.STACS.2026.23.pdf
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ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Fixed parameter tractability
Keywords
  • Minimum Circuit Size Problem
  • Circuit Lower Bounds
  • Exponential Time Hypothesis

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Abstract

The Minimum Circuit Size Problem for Partial Functions (MCSP^*) is hard assuming the Exponential Time Hypothesis (ETH) (Ilango, 2020). This breakthrough result leveraged a characterization of the optimal {∧, ∨, ¬} circuits for n-bit OR (OR_n) and a reduction from the partial f-Simple Extension Problem where f = OR_n. It remains open to extend that reduction to show ETH-hardness of total MCSP. However, Ilango observed that the total f-Simple Extension Problem is easy whenever f is computed by read-once formulas (like OR_n). Therefore, extending Ilango’s proof to total MCSP would require replacing OR_n with a more complex but similarly well-understood Boolean function.
This work shows that the f-Simple Extension problem remains easy when f is the next natural candidate: XOR_n. We first develop a fixed-parameter tractable algorithm for the f-Simple Extension Problem that is efficient whenever the optimal circuits for f are (1) linear in size, (2) polynomially "few" and efficiently enumerable in the truth-table size (up to isomorphism and permutation of inputs), and (3) all have constant bounded fan-out. XOR_n satisfies all three of these conditions. When ¬ gates count towards circuit size, optimal XOR_n circuits are binary trees of n-1 subcircuits computing (¬)XOR₂ (Kombarov, 2011). We extend this characterization when ¬ gates do not contribute the circuit size. Thus, the XOR-Simple Extension Problem is in polynomial time under both measures of circuit complexity.
We conclude by discussing conjectures about the complexity of the f-Simple Extension problem for each explicit function f with known and unrestricted circuit lower bounds over the DeMorgan basis. Examining the conditions under which our Simple Extension Solver is efficient, we argue that multiplexer functions (MUX) are the most promising candidate for ETH-hardness of a Simple Extension Problem, towards proving ETH-hardness of total MCSP.

Cite As Get BibTex

Marco Carmosino, Ngu Dang, and Tim Jackman. Simple Circuit Extensions for XOR in PTIME. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.23

Author Details

Marco Carmosino
  • MIT-IBM Watson AI Lab, Cambridge, MA, USA
Ngu Dang
  • Boston University, MA, USA
Tim Jackman
  • Boston University, MA, USA

Acknowledgements

We would like to thank Rahul Ilango for many invaluable discussions.

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